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Semitopological groups, Bouziad spaces and topological groups
Topology and its Applications 160 (2013) 2038–2048
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Topology and its Applications www.elsevier.com/locate/topol
Semitopological groups, Bouziad spaces and topological groups Warren B. Moors ∗ Department of Mathematics, The University of Auckland, Private Bag 92019, Auckland, New Zealand
a r t i c l e
i n f o
Article history: Received 24 February 2013 Received in revised form 14 August 2013 Accepted 15 August 2013
a b s t r a c t A semitopological group (topological group) is a group endowed with a topology for which multiplication is separately continuous (multiplication is jointly continuous and inversion is continuous). In this paper we use topological games to show that many semitopological groups are in fact topological groups. © 2013 Elsevier B.V. All rights reserved.
MSC: primary 22A20, 91A44 secondary 54E18, 54H11, 54H15 Keywords: Topological group Semitopological group Topological games Strongly Baire spaces
1. Introduction A semitopological group (topological group) is a group endowed with a topology for which multiplication is separately continuous (multiplication is jointly continuous and inversion is continuous). Recall that a function f : X × Y → Z that maps from a product of topological spaces X and Y into a topological space Z is said to be jointly continuous at a point (x, y) ∈ X × Y if for each neighbourhood W of f (x, y) there exists a pair of neighbourhood U of x and V of y such that f (U × V ) ⊆ W . If f is jointly continuous at each point of X × Y then we say that f is jointly continuous on X × Y . A related but weaker notion of continuity is the following. A function g : X × Y → Z that maps from a product of topological spaces X and Y into a topological space Z is said to be separately continuous on X × Y if for each x0 ∈ X and y0 ∈ Y the functions y → g(x0 , y) and x → g(x, y0 ) are both continuous on Y and X respectively. Ever since [22] there has been continued interest in determining topological properties of a semitopological group that are sufficient to ensure that it is a topological group. There have been many significant contributions to this area, see [1–10,12–14,17,18,20,22–32] to name but a few. Just about all of these results require the semitopological group to be regular (i.e., every closed subset and every point outside of this set, * Tel.: +64 9 3737 599x84746. E-mail address: [email protected]. 0166-8641/$ – see front matter © 2013 Elsevier B.V. All rights reserved. http://dx.doi.org/10.1016/j.topol.2013.08.008
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can be separated by disjoint open sets) and Baire (i.e., the intersection of any countable family of dense open sets is dense), and satisfy some additional completeness properties. This paper is no exception. However, in this paper we also require some additional notions from topology and game theory. We shall say that a subset Y of a topological space (X, τ ) is bounded in X if for each decreasing sequence of open sets (Un )n∈N in X, n∈N Un = ∅, whenever Un ∩ Y = ∅ for each n ∈ N. The game that we shall consider involves two players which we will call α and β. The “field/court” that the game is played on is a fixed topological space (X, τ ) with a fixed dense subset D. The name of the game is the GB (D)-game. After naming the game we need to describe how to “play” the GB (D)-game. The player labeled β starts the game every time (life is not always fair). For his/her first move the player β must select a pair (B1 , b1 ) consisting of a nonempty open subset B1 ⊆ X and a point b1 ∈ D. Next, α gets a turn. For α’s first move he/she must select a nonempty open subset A1 of B1 . This ends the first round of the game. In the second round, β goes first again and selects a pair (B2 , b2 ) consisting of a nonempty open subset B2 ⊆ A1 and a point b2 ∈ A1 ∩ D. Player α then gets to respond by choosing a nonempty open subset A2 of B2 . This ends the second round of the game. In general, after α and β have played the first n-rounds of the GB (D)-game, β will have selected pairs (B1 , b1 ), (B2 , b2 ), . . . , (Bn , bn ) consisting of nonempty open sets B1 , B2 , . . . , Bn and points b1 , b2 , . . . , bn in D and α will have selected nonempty open subsets A1 , A2 , . . . , An such that An ⊆ Bn ⊆ An−1 ⊆ Bn−1 ⊆ · · · ⊆ A2 ⊆ B2 ⊆ A1 ⊆ B1 , and bk+1 ∈ Ak ∩ D for all 1 k < n. At the start of the (n + 1)-round of the game, β goes first (again!) and selects a pair (Bn+1 , bn+1 ) consisting of a nonempty open subset Bn+1 of An and a point bn+1 ∈ An ∩ D. As with the previous n-rounds, the player α gets to respond to this move by selecting a nonempty open subset An+1 of Bn+1 . Continuing this procedure indefinitely (i.e., continuing on forever) the players α and β produce an infinite sequence (An , (Bn , bn ))n∈N called a play of the GB (D)-game. A partial play ((Ak , (Bk , bk )): 1 k n) of the GB (D)-game consists of the first n-moves of the GB (D)-game. As with any game, we need to specify a rule to determine who wins (otherwise, it is a very boring game). We shall declare that α wins a play (An , (Bn , bn ))n∈N of the GB (D)-game if: for each decreasing sequence / Un } is finite, for every n ∈ N. (Un )n∈N of open subsets of X, n∈N Un = ∅, whenever {k ∈ N: bk ∈ If α does not win a play of the GB (D)-game then we declare that β wins that play of the GB (D)-game. So every play is won by either α or β and no play is won by both players. Note that if α wins a play (An , (Bn , bn ))n∈N of the GB (D)-game then n∈N An = ∅. Continuing further into game theory we need to introduce the notion of a strategy. By a strategy t for the player β we mean a ‘rule’ that specifies each move of the player β in every possible situation. More precisely, a strategy t := (tn : n ∈ N) for β is an inductively defined sequence of τ × D-valued functions. The domain of t1 is the sequence of length zero, denoted by ∅. That is, Dom(t1 ) = {∅} and t1 (∅) ∈ (τ \ {∅}) × D. If t1 , t2 , . . . , tk have been defined then the domain of tk+1 is:
k (A1 , A2 , . . . , Ak ) ∈ τ \ {∅} : (A1 , A2 , . . . , Ak−1 ) ∈ Dom(tk ) and Ak ⊆ Bk , where (Bk , bk ) := tk (A1 , A2 , . . . , Ak−1 ) .
For each (A1 , A2 , . . . , Ak ) ∈ Dom(tk+1 ), tk+1 (A1 , A2 , . . . , Ak ) := (Bk+1 , bk+1 ) ∈ (τ \ {∅}) × D is defined so that Bk+1 ⊆ Ak and bk+1 ∈ Ak ∩ D. A partial t-play is a finite sequence (A1 , A2 , . . . , An−1 ) such that (A1 , A2 , . . . , An−1 ) ∈ Dom(tn ). A t-play is an infinite sequence (An )n∈N such that for each n ∈ N, (A1 , A2 , . . . , An−1 ) is a partial t-play. A strategy t := (tn : n ∈ N) for the player β is called a winning strategy if each play of the form: (An , tn (A1 , . . . , An−1 ))n∈N is won by β. We will call a topological space (X, τ ) a Bouziad space if it is
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regular and there exists a dense subset D of X such that the player β does not have a winning strategy in the GB (D)-game played on X. It follows from [29, Theorem 1] that each Bouziad space is in fact a Baire space. In this paper we will also be interested in another closely related game. This game, denoted GSB (D), is identical to the GB (D)-game, except for the definition of a win. In the GSB (D)-game we say that α wins a play (An , (Bn , bn ))n∈N of the GSB (D)-game if (i) there exists a subspace S of X and a subsequence (bnk )k∈N of (bn )n∈N , contained in S, such that for each decreasing sequence (Un )n∈N of open subsets of S, S / Un } is finite, for every n ∈ N and (ii) every subspace of Cp (S), n∈N Un = ∅, whenever {k ∈ N: bnk ∈ that is bounded in Cp (S), has a compact closure. We will call a topological space (X, τ ) a strong Bouziad space if it is completely regular and there exists a dense subset D of X such that the player β does not have a winning strategy in the GSB (D)-game played on X. It is easy to show that every strongly Bouziad space is a Bouziad space, and hence a Baire space. On the other hand, if X is completely regular and has the property that every subset of Cp (X), that is bounded in Cp (X), has a compact closure, then X is a Bouziad space if, and only if, it is a strongly Bouziad space. 2. Feeble continuity of multiplication Lemma 1. Let (G, ·, τ ) be a semitopological group. If (G, τ ) is a Bouziad space then for each pair of open neighbourhoods U and W of identity element e ∈ G there exists a nonempty open subset V of U such that V −1 ⊆ W · W · W . Proof. Suppose, in order to obtain a contradiction, that there exists a pair of open neighbourhoods U and W of e ∈ G such that for each nonempty open subset V of U , V −1 W · W · W . From this it follows that for each nonempty open subset V of U and each dense subset D of V there exists a point x ∈ V ∩ D such / W · W , because otherwise, that x−1 ∈ −1 −1 V −1 ⊆ V ∩ D ⊆ W · V ∩ D ⊆ W · W · W ⊆ W · W · W.
Recall that for any nonempty subset A of a semitopological group (H, ·, τ ) and any open neighbourhood W of the identity element e ∈ H, (A)−1 ⊆ W · A−1 . Now, let D be any dense subset of G such that β does not have a winning strategy in the GB (D)-game played on G. We will define a (necessarily non-winning) strategy t := (tn : n ∈ N) for β in the GB (D)-game played on G, but first we set, for notational reasons, A0 := U and b0 := e. −1 Step 1. Choose b1 ∈ A0 ∩D so that (b−1 = b−1 / W · W . Then choose U1 to be any open neighbourhood 0 ·b1 ) 1 ∈ of e, contained in U ∩ W , such that b1 · U1 ⊆ A0 . Then define t1 (∅) := (b1 · U1 , b1 ). Now, suppose that bj , Uj and tj (A1 , . . . , Aj−1 ) have been defined for each 1 j n so that: −1 (i) bj ∈ Aj−1 ∩ D and (b−1 ∈ / W · W; j−1 · bj ) (ii) Uj is an open neighbourhood of e, contained in U ∩ W , such that bj · Uj ⊆ Aj−1 ; (iii) tj (A1 , . . . , Aj−1 ) := (bj · Uj , bj ).
Step n + 1. Suppose that An is a nonempty open subset of bn · Un . That is, suppose that An is the n-th −1 move of the player α. Choose bn+1 ∈ An ∩ D so that (b−1 ∈ / W · W . Note that this is possible since n · bn+1 ) −1 −1 bn · (An ∩ D) is a dense subset of bn · An and −1 b−1 n · An ⊆ bn · (bn · Un ) = Un ⊆ U.
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Then choose Un+1 to be any neighbourhood of e, contained in U ∩ W , such that bn+1 · Un+1 ⊆ An . Finally, define tn+1 (A1 , . . . , An ) := (bn+1 · Un+1 , bn+1 ). Note that: −1 (i) bn+1 ∈ An ∩ D and (b−1 ∈ / W · W; n · bn+1 ) (ii) Un+1 is an open neighbourhood of e, contained in U ∩ W , such that bn+1 · Un+1 ⊆ An ; (iii) tn+1 (A1 , . . . , An ) := (bn+1 · Un+1 , bn+1 ).
This completes the definition of t. Since t is not a winning strategy for β there exists a play (An , tn (A1 , . . . , An−1 ))n∈N where α wins. Now there exists a k ∈ N such that bk ∈ ( n∈N Bn ) · W , because otherwise,
{bn : k + 1 n} ⊆ Bk \ Bn · W
for all k ∈ N.
n∈N
In which case,
∅ = Bk \ Bn · W ⊆ G \ Bn · W, k∈N
n∈N
n∈N
but on the other hand,
Bk \ Bn · W ⊆ Bn = Bn ⊆ Bn · W, n∈N
k∈N
n∈N
n∈N
n∈N
which is impossible. Therefore, we may indeed choose k ∈ N so that bk ∈
Bn
· W ⊆ Ak+1 · W ⊆ bk+1 · Uk+1 · W ⊆ bk+1 · W · W = bk+1 · W · W .
n∈N −1 Therefore, (b−1 = b−1 k · bk+1 ) k+1 · bk ∈ W · W . However, this contradicts the way bk+1 was chosen. This completes the proof. 2
If f : (X, τ ) → (Y, τ ) is a surjection acting between topological spaces (X, τ ) and (Y, τ ) then we say that f is feebly continuous on X if for each nonempty open subset V of Y , int[f −1 (V )] = ∅, [10,15]. Proposition 1. Let (G, ·, τ ) be a semitopological group. If multiplication, (h, g) → h · g, is feebly continuous on G × G then for each nonempty open subset U of G and n ∈ N there exist a point x in U and an open neighbourhood V of the identity element e ∈ G such that: x · V · V · V · · · V ⊆ U n-times
and
V · V · V · · · V · x ⊆ U. n-times
Proof. The proof of this follows from a simple induction argument and the fact that for each g ∈ G, both {g · U : U is a neighbourhood of e} and {U · g: U is a neighbourhood of e} are local bases for τ at the point g ∈ G. 2 Lemma 2. Let (G, ·, τ ) be a semitopological group. If (G, τ ) is a Bouziad space and the multiplication operation (h, g) → h · g is feebly continuous on G × G then the multiplication operation, (h, g) → h · g, is jointly continuous on G × G.
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Proof. Since (G, ·, τ ) is a semitopological group it is sufficient to show that multiplication is jointly continuous at (e, e). So, in order to obtain a contradiction, we will assume that multiplication is not jointly continuous at (e, e). Therefore, by the regularity of (G, τ ), there exists an open neighbourhood W of e so that for every neighbourhood U of e, U · U W . Since (G, τ ) is a Bouziad space there exists a dense subset DG of G such that β does not possess a winning strategy in the GB (DG )-game played on G. We will now inductively define a (necessarily non-winning) strategy t := (tn : n ∈ N) for the player β in the GB (DG )-game played on G. Step 1. We may choose a point x ∈ A0 := G and an open neighbourhood U of e ∈ G such that x · U ⊆ x · U · U · U ⊆ x · U · U · U ⊆ A0 . Next, we may pick y, z ∈ U such that y ·z ∈ / W (i.e., y ∈ / W · z −1 and so U \ (W · z −1 ) = ∅). By Proposition 1 −1 we may select a point y ∈ U \ (W · z ) and an open neighbourhood V of e, contained in U , such that V · V · V · V · y ⊆ U \ W · z −1 .
Then, (V ·V ·V ·x−1 )·(x·V )·y ·z ∩ W = ∅ and so (V · V · V ·x−1 )·(x·V )·y ·z ∩W = ∅. By Lemma 1 there exists a nonempty open subset B1 of x · V ⊆ x · U ⊆ x · U · U · U ⊆ A0 such that (B1 )−1 ⊆ V · V · V · x−1 . Thus, (B1 )−1 · B1 · y · z ∩ W = ∅. Choose b1 ∈ B1 · y · z ∩ DG ⊆ B1 · U · U ⊆ x · V · U · U ⊆ x · U · U · U ⊆ A0 .
Then define t1 (∅) := (B1 , b1 ) and U1 := B1 · y · z. Note that: (B1 )−1 · U1 ∩ W = ∅ — (∗1 ). Now suppose that (Bj , bj ), Uj and tj (A1 , . . . , Aj−1 ) have been defined for each 1 j n so that: (i) Uj is an open subset of Aj−1 and bj ∈ Uj ∩ DG ; (ii) Bj ⊆ Aj−1 and (Bj )−1 · Uj ∩ W = ∅ — (∗j ); (iii) tj (A1 , . . . , Aj−1 ) := (Bj , bj ). Step n + 1. Suppose that An is a nonempty open subset of Bn . That is, suppose that An is the n-th move of the player α. By Proposition 1 we may choose a point x ∈ An and an open neighbourhood U of e ∈ G such that x · U ⊆ x · U · U · U ⊆ x · U · U · U ⊆ An . Next, we may pick y, z ∈ U such that y · z ∈ / W (i.e., y ∈ / W · z −1 and so U \ (W · z −1 ) = ∅). Again by −1 Proposition 1 we may select a point y ∈ U \ (W · z ) and an open neighbourhood V of e, contained in U , such that V · V · V · V · y ⊆ U \ W · z −1 .
Then, (V · V · V · x−1 ) · (x · V ) · y · z ∩ W = ∅ and so (V · V · V · x−1 ) · (x · V ) · y · z ∩ W = ∅. By Lemma 1 there exists a nonempty open subset Bn+1 of x · V ⊆ x · U ⊆ An such that (Bn+1 )−1 ⊆ V · V · V · x−1 . Thus, (Bn+1 )−1 · Bn+1 · y · z ∩ W = ∅. Choose bn+1 ∈ Bn+1 · y · z ∩ DG ⊆ Bn+1 · U · U ⊆ x · V · U · U ⊆ x · U · U · U ⊆ An .
Then define tn+1 (A1 , . . . , An ) := (Bn+1 , bn+1 ) and Un+1 := Bn+1 · y · z. Note that: Bn+1 ⊆ An and (Bn+1 )−1 · Un+1 ∩ W = ∅ — (∗n+1 ).
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This completes the definition of t. Since t is not a winning strategy for β there exists a play (An , tn (A1 , . . . , An−1 ))n∈N where α wins. Note that since {bn : n k} ⊆ ( nk Un ) for all k ∈ N, ∅ =
k∈N
Un
⊆
nk
k∈N
Ak−1 =
k∈N
Ak =
k∈N
Bk =
Bn .
n∈N
Let b∞ ∈ k∈N ( nk Un ). Fix n ∈ N, then by Eq. (∗n ), b−1 ∞ ·Un ∩W = ∅, or equivalently, Un ∩(b∞ ·W ) = ∅. Hence, k∈N ( nk Un ) ∩ (b∞ · W ) = ∅. However, this contradicts the fact that b∞ ∈ k∈N ( nk Un ) and W is an open neighbourhood of e. Hence the multiplication operation on G is jointly continuous. 2
If f : (X, τ ) → (Y, τ ) is a function acting between topological spaces (X, τ ) and (Y, τ ) and x ∈ X then we say that f is quasi-continuous at x if for each neighbourhood W of f (x) and neighbourhood U of x there exists a nonempty open subset V ⊆ U such that f (V ) ⊆ W , [16]. Theorem 1. If (G, ·, τ ) is a semitopological group and (G, τ ) is a Bouziad space then (G, ·, τ ) is a topological group if (and only if ) the multiplication operation (h, g) → h · g is feebly continuous on G × G. Proof. From Lemma 2 we know that the multiplication operation on G is jointly continuous. Therefore, by Lemma 1, we see that inversion is quasi-continuous at e. The result now follows from [17, Lemma 4] where it is shown that each semitopological group with jointly continuous multiplication and inversion that is quasi-continuous at the identity element is a topological group. 2 Hence in the realm of Bouziad spaces, the problem of determining when a semitopological group is a topological group reduces to the problem of determining when a semitopological group has feebly continuous multiplication. 3. Strong quasi-continuity of multiplication Let X, Y and Z be topological spaces. We will say that a function f : X × Y → Z is strongly quasicontinuous, with respect to the second variable at (x, y) ∈ X × Y , if for each neighbourhood W of f (x, y) and each neighbourhood U of x there exists a nonempty open subset U ⊆ U and a neighbourhood V of y such that f (U × V ) ⊆ W , [25]. Remark 1. It follows from Proposition 1 that the multiplication operation on a semitopological group (G, ·, τ ) is feebly continuous on G × G if, and only if, it is strongly quasi-continuous, with respect to the second variable, at the point (e, e) ∈ G × G. If (X, τ ) is a topological space and a ∈ X then we shall denote by N (a) the family of all neighbourhoods of a. For any point a in a topological space (X, τ ) and any dense subset D of X we can consider the following two player topological game, called the Gp (a, D)-game. To define this game we must first specify the “rules” and then also specify the definition of a “win”. The moves of the player α are simple. He/she must always select a neighbourhood of the point a. However, the moves of the player β may depend upon the previous move of α. Specifically, for his/her first move β may select any point b1 ∈ D. For α’s first move, as mentioned earlier, α must select a neighbourhood A1 of a. Now, for β’s second move he/she must select a point b2 ∈ A1 ∩ D. For α’s second move he/she is entitled to select any neighbourhood A2 of a. In general, if α has chosen An ∈ N (a) as his/her n-th move of the Gp (a, D)-game then β is obliged to choose a point bn+1 ∈ An ∩ D. The response of α is then simply to
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choose any neighbourhood An+1 of a. Continuing in this fashion indefinitely, the players α and β produce a sequence ((bn , An ))n∈N of ordered pairs with bn+1 ∈ An ∩ D ⊆ An ∈ N (a) for all n ∈ N, called a play of the Gp (a, D)-game. A partial play ((bk , Ak ): 1 k n) of the Gp (a, D)-game consists of the first n moves of a play of the Gp (a, D)-game. We shall declare α the winner of a play ((bn , An ))n∈N of the Gp (a, D)-game if: for each decreasing sequence (Un )n∈N of open subsets of X, n∈N Un = ∅, whenever {k ∈ N: bk ∈ / Un } is finite, for every n ∈ N, otherwise, β is declared the winner. Note that if n∈N {bk : k n} = ∅ then α wins the corresponding play ((bn , An ))n∈N . A strategy for the player α is a rule that specifies his/her moves in every possible situation that can occur. More precisely, a strategy for α is an inductively defined sequence of functions t := (tn : n ∈ N). The domain of t1 is D1 and for each (b1 ) ∈ D1 , t1 (b1 ) ∈ N (a), i.e., ((b1 , t1 (b1 ))) is a partial play. Inductively, if t1 , t2 , . . . , tn have been defined then the domain of tn+1 is defined to be
(b1 , b2 , . . . , bn+1 ) ∈ Dn+1 : (b1 , b2 , . . . , bn ) ∈ Dom(tn ) and bn+1 ∈ tn (b1 , b2 , . . . , bn ) ∩ D .
For each (b1 , b2 , . . . , bn+1 ) ∈ Dom(tn+1 ), tn+1 (b1 , b2 , . . . , bn+1 ) ∈ N (a). Equivalently, for each (b1 , b2 , . . . , bn+1 ) ∈ Dom(tn+1 ), ((bk , tk (b1 , . . . , bk )): 1 k n + 1) is a partial play. A partial t-play is a finite sequence (b1 , b2 , . . . , bn ) ∈ Dn such that (b1 , b2 , . . . , bn ) ∈ Dom(tn ) or, equivalently, if bk+1 ∈ tk (b1 , b2 , . . . , bk ) ∩ D for all 1 k < n. A t-play is an infinite sequence (bn )n∈N such that for each n ∈ N, (b1 , b2 , . . . , bn ) is a partial t-play. A strategy t := (tn : n ∈ N) for the player α is said to be a winning strategy if each play of the form: (bn , tn (b1 , b2 , . . . , bn ))n∈N is won by α. ∗ We shall call a point a a nearly qD -point if the player α has a winning strategy in the Gp (a, D)-game played on X. For more information on topological games, see [11,19]. ∗ Lemma 3. Suppose that D is a dense subset of a topological space (X, τ ) and a is a nearly qD -point of X, then the player α possesses a strategy s := (sn : n ∈ N) in the Gp (a, D)-game such that every s-play is bounded in X.
Proof. Let t := (tn : n ∈ N) be a winning strategy for the player α in the Gp (a, D)-game. We shall inductively define a new strategy s := (sn : n ∈ N) for the player α. Step 1. Let (b1 ) be a partial s-play, i.e., b1 ∈ D, define s1 (b1 ) := t1 (b1 ). Now, suppose that s1 , s2 , . . . , sk have been defined so that for each 1 i k: (i) if (bn1 , . . . , bnm ) is a subsequence of (b1 , . . . , bi ) ∈ Dom(si ), then (bn1 , . . . , bnm ) ∈ Dom(tm ); (ii) if (b1 , . . . , bi ) ∈ Dom(si ) then si (b1 , . . . , bi ) =
tm (bn1 , . . . , bnm ): (bn1 , . . . , bnm ) is a subsequence of (b1 , . . . , bi ) .
Step k + 1. Suppose that (b1 , . . . , bk+1 ) is a partial s-play, i.e., (b1 , . . . , bk ) ∈ Dom(sk ) and bk+1 ∈ sk (b1 , . . . , bk ) ∩ D. Let (bn1 , . . . , bnm ) be any subsequence of (b1 , . . . , bk+1 ) and consider the following three cases: (i) if nm < k + 1, then (bn1 , . . . , bnm ) is a subsequence of (b1 , . . . , bk ) and so by the induction hypothesis (bn1 , . . . , bnm ) ∈ Dom(tm ); (ii) if nm = k + 1 and m > 1, then (bn1 , . . . , bnm−1 ) is a subsequence of (b1 , . . . , bk ) and so by the induction hypothesis (bn1 , . . . , bnm−1 ) ∈ Dom(tm−1 ). On the other hand, bnm = bk+1 ∈ sk (b1 , . . . , bk ) ∩ D ⊆ tm−1 (bn1 , . . . , bnm−1 ) ∩ D and so (bn1 , . . . , bnm ) ∈ Dom(tm ); (iii) if nm = k + 1 and m = 1, i.e., if n1 = k + 1 then (bn1 ) = (bk+1 ) ∈ Dom(t1 ).
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In all three cases, (bn1 , . . . , bnm ) ∈ Dom(tm ). Define sk+1 (b1 , . . . , bk+1 ) :=
tm (bn1 , . . . , bnm ): (bn1 , . . . , bnm ) is a subsequence of (b1 , . . . , bk+1 ) .
This completes the definition of s. Now suppose that (bn )n∈N is an s-play, we need to show that {bn : n ∈ N} is bounded in X. To this end, let (Un )n∈N be a decreasing sequence of open sets such that Un ∩{bk : k ∈ N} = ∅ for each n ∈ N. If for some n ∈ N, {k ∈ N: bk ∈ Un } is finite then n∈N Un = ∅. So we may suppose that for each n ∈ N, {k ∈ N: bk ∈ Un } is infinite. For each n ∈ N, let Jn := {k ∈ N: bk ∈ Un }. Then (Jn )n∈N is a decreasing sequence of infinite subsets of N. Hence there exists a strictly increasing sequence (nk )k∈N of the natural numbers such that nk ∈ Jk for all k ∈ N. We claim that (bnk )k∈N is a t-play. To show this we must show that for each m ∈ N, (bn1 , . . . , bnm ) ∈ Dom(tm ). However, (bn1 , . . . , bnm ) is a subsequence of (b1 , b2 , . . . , bnm ) ∈ Dom(snm ). Therefore, by the construction of the strategy s, (bn1 , . . . , bnm ) ∈ Dom(tm ). Hence (bnk )k∈N is a t-play. Now, |{k ∈ N: bnk ∈ / Un }| < n for each n ∈ N, therefore, n∈N Un = ∅. This shows that {bk : k ∈ N} is bounded in X. 2 Variations of the following result are well known, see [6,7,12,17,20]. Lemma 4. Let X be a strongly Bouziad space, Y be a topological space and Z be a completely regular space. If f : X × Y → Z is a separately continuous function and D is a dense subset of Y , then for each nearly ∗ qD -point y ∗ ∈ Y the function f is strongly quasi-continuous, with respect to the second variable, at each point of X × {y ∗ }. Proof. Let DX be any dense subset of X such that β does not have a winning strategy in the GSB (DX )-game played on X. (Note: such a dense subset is guaranteed by the fact that X is a strong Bouziad space.) We need to show that f is strongly quasi-continuous, with respect to the second variable, at each point (x∗ , y ∗ ) ∈ X × {y ∗ }. So in order to obtain a contradiction let us assume that f is not strongly quasi-continuous, with respect to the second variable, at some point (x∗ , y ∗ ) ∈ X × {y ∗ }. Then there exist open neighbourhoods W of f (x∗ , y ∗ ) and U of x∗ so that f (U × V ) W for each nonempty open subset U of U and each neighbourhood V of y0 . By the complete regularity of Z there exists a continuous function g : Z → [0, 1] such that g(f (x∗ , y ∗ )) = 1 and g(Z \ W ) = {0}. Let W := {z ∈ Z: g(z) > 3/4} ⊆ W . Note that by possibly making U smaller we may assume that f (x, y ∗ ) ∈ W for all x ∈ U . We will now inductively define a strategy t := (tn : n ∈ N) for the player β in the GSB (DX )-game played on X. However, we shall first: (a) let s := (sn : n ∈ N) be a strategy for the player α in the Gp (y ∗ , D)-game such that every s-play is bounded in Y . Note that by Lemma 3 such a strategy exists; (b) set (for notational reasons) x0 := x∗ , A0 := U and V0 := Y and let y0 be any element of D. Step 1. Select (x1 , y1 ) ∈ X × Y and an open set V1 and t1 (∅) so that: (i) y ∗ ∈ V1 := {y ∈ V0 ∩ s1 (y0 ): f (x0 , y) ∈ W }; (ii) (x1 , y1 ) ∈ (A0 ∩ DX ) × (V1 ∩ D) and f (x1 , y1 ) ∈ / W; (iii) t1 (∅) := (B1 , x1 ) where, B1 := {x ∈ A0 : f (x, y1 ) ∈ / W }. Now suppose that (xj , yj ), Vj and tj have been defined for each 1 j n so that for each 1 j n (i) y ∗ ∈ Vj := {y ∈ Vj−1 ∩ sj (y0 , . . . , yj−1 ): f (xj−1 , y) ∈ W }; (ii) (xj , yj ) ∈ (Aj−1 ∩ DX ) × (Vj ∩ D) and f (xj , yj ) ∈ / W; (iii) tj (A1 , . . . , Aj−1 ) := (Bj , xj ), where Bj := {x ∈ Aj−1 : f (x, yj ) ∈ / W }.
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Step n + 1. Suppose that An is a nonempty open subset of Bn . That is, suppose that An is the n-th move of the player α. Select (xn+1 , yn+1 ) ∈ X × Y and open set Vn+1 and tn+1 (A1 , . . . , An ) so that: (i) y ∗ ∈ Vn+1 := {y ∈ Vn ∩ sn+1 (y0 , . . . , yn ): f (xn , y) ∈ W }; (ii) (xn+1 , yn+1 ) ∈ (An ∩ DX ) × (Vn+1 ∩ D) and f (xn+1 , yn+1 ) ∈ / W; (iii) tn+1 (A1 , . . . , An ) := (Bn+1 , xn+1 ), where Bn+1 := {x ∈ An : f (x, yn+1 ) ∈ / W }. This completes the definition of t := (tn : n ∈ N). Now since t is not a winning strategy for the player β in the GSB (DX )-game there exists a play (An , tn (A1 , . . . , An−1 ))n∈N where α wins. Therefore, there exists a subspace S ⊆ X and a subsequence (xnk )k∈N of (xn )n∈N , contained in S, such that for each decreasing / Un } is finite, for every sequence (Un )n∈N of open subsets of S, n∈N Un S = ∅, whenever {k ∈ N: xnk ∈ n ∈ N. Define ϕ : Y → Cp (S) – [the continuous real-valued functions defined on S, endowed with the topology of pointwise convergence on S] by ϕ(y)(s) := (g ◦ f )(s, y)
for all s ∈ S.
Then ϕ is well defined and continuous on Y . Now, since (yn )n∈N is an s-play, {yn : n ∈ N} is bounded in Y and so {ϕ(ym ): m ∈ N} is bounded in Cp (S). Thus, by assumption, {ϕ(ym ): m ∈ N}τp is a compact subspace of Cp (S). Hence the sequence (ϕ(yn ))n∈N has a cluster point h ∈ C(S). Now, for each fixed k ∈ N, f (xnk , yi ) ∈ f {xnk } × Vi ⊆ f {xnk } × Vnk +1 ⊆ W
for all i > nk , since yi ∈ Vi for all i ∈ N. Therefore, ϕ(yi )(xnk ) ∈ (3/4, 1] for all i > nk and so h(xnk ) ∈ [3/4, 1] ⊆ (2/3, 1] for all k ∈ N. Since h is continuous, for every k ∈ N there exists a relatively open subset Uk of S such that xnk ∈ Uk ⊆ Ank −1 and h(Uk ) ⊆ (2/3, 1]. Hence the set
Ui S = ∅.
k∈N ik
[Here, X S denotes the closure of a subset X of S with respect to the relative topology on S.] Let x∞ ∈ k∈N ik Ui S ⊆ S. Then h(x∞ ) ∈ [2/3, 1]. On the other hand, if we again fix k ∈ N then f Ui × {yk } ⊆ f Ani −1 × {yk } ⊆ f Ai−1 × {yk } ⊆ f Ak × {yk } ⊆ Z \ W for all i > k. Therefore, f ( i>k Ui S × {yk }) ⊆ Z \ W for each k ∈ N and so f (x∞ , yk ) ∈ Z \ W for each k ∈ N; which implies that h(x∞ ) = 0. This however, contradicts our earlier conclusion that h(x∞ ) ∈ [2/3, 1]. Hence f is strongly quasi-continuous, with respect to the second variable, at (x∗ , y ∗ ). 2
Corollary 1. Suppose that (G, ·, τ ) is a semitopological group such that (G, τ ) is a strong Bouziad space. ∗ -point, for some dense subset Then (G, ·, τ ) is a topological group, provided (G, τ ) has at least one nearly qD D of G. Proof. Since for each g ∈ G, x → g · x is a homeomorphism on G, we may assume that the identity element ∗ -point, for some dense subset D of G. Then by Lemma 4 the multiplication operation e ∈ G is a nearly qD on G is strongly quasi-continuous, with respect to the second variable, at the point (e, e) ∈ G × G. Hence, by Remark 1 the multiplication operation on G is feebly continuous. The result now follows from Theorem 1 since every strongly Bouziad space is a Bouziad space. 2
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If (X, τ ) is a topological space and a ∈ X, then we call the point a, a q-point if there exists a sequence of neighbourhoods (Un )n∈N of a such that every sequence (xn )n∈N in X, with xn ∈ Un for all n ∈ N, has a cluster-point in X. Corollary 2. Suppose that (G, ·, τ ) is a semitopological group such that (G, τ ) is a Bouziad space. Then (G, ·, τ ) is a topological group, provided (G, τ ) has at least one q-point. Proof. It is well known that if K is a q-space, then every subset of Cp (K), that is bounded in Cp (K), has a compact closure. Hence, by the comment at the end of the introduction to this paper, we see that (G, τ ) is ∗ -point of G. Hence this a strong Bouziad space. Moreover, it is clear that every q-point of G is a nearly qG result follows from Corollary 1. 2 Corollary 3. ([28]) Suppose that (G, ·, τ ) is a semitopological group such that (G, τ ) is pseudo-compact. Then (G, ·, τ ) is a topological group, provided every subset of Cp (G), that is bounded in Cp (G), has a compact closure. Proof. From the comment at the end of the introduction to this paper, it follows that (G, τ ) is a strong ∗ -point of G. So the Bouziad space. Furthermore, since G is pseudo-compact, every point of G is a nearly qG result follows from Corollary 1. 2 Corollary 4. ([21]) If (G, ·, τ ) is a semitopological group such that (G, τ ) is homeomorphic to a product of Čech-complete spaces, then (G, ·, τ ) is a topological group. Proof. It is easy to see that every nearly strongly Baire space (see, [21] for the definition) is a strong Bouziad ∗ -point. Now, in [21] it is space and that every nearly qD -point (see [21] for the definition) is a nearly qD shown that every space that is homeomorphic to a product of Čech-complete spaces is a nearly strongly Baire space with at least one nearly qD -point. Hence the result follows from Corollary 1. 2 Corollary 5. ([17]) If (G, ·, τ ) is a semitopological group such that (G, τ ) is a completely regular strongly Baire space, then (G, ·, τ ) is a topological group. Proof. It is easy to see that every completely regular, strongly Baire space (see [17] for the definition) is a ∗ -point. The result then follows from Corollary 1. 2 strong Bouziad space and possesses at least one nearly qD Acknowledgement After this paper went to press the author became aware of the paper “Continuity of the inverse” located at: http://gtopology.math.msu.su/node/175 by Evgenii Reznichenko where the author proves a theorem similar to Theorem 1. References [1] A.V. Arhangel’skii, M.M. Choban, P.S. Kenderov, Topological games and topologies on groups, Math. Maced. 8 (2010) 1–19. [2] A.V. Arhangel’skii, M.M. Choban, P.S. Kenderov, Topological games and continuity of group operations, Topol. Appl. 157 (2010) 2542–2552. [3] A.V. Arhangel’skii, E.A. Reznichenko, Paratopological and semitopological groups versus topological groups, Topol. Appl. 151 (2005) 107–119. [4] A.V. Arhangel’skii, M.G. Tkachenko, Topological Groups and Related Structures, Atlantis Press, Amsterdam/Paris, 2008. [5] A. Bareche, A. Bouziad, Some results on separate and joint continuity, Topol. Appl. 157 (2010) 327–335. [6] A. Bouziad, The Ellis theorem and continuity in groups, Topol. Appl. 50 (1993) 73–80. [7] A. Bouziad, Continuity of separately continuous group actions in p-spaces, Topol. Appl. 71 (1996) 119–124.
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[8] A. Bouziad, Every Čech-analytic Baire semi-topological group is a topological group, Proc. Am. Math. Soc. 124 (1996) 953–959. [9] N. Brand, Another note on the continuity of the inverse, Arch. Math. 39 (1982) 241–245. [10] J. Cao, R. Drozdowski, Z. Piotrowski, Weak continuity properties of topologized groups, Czechoslov. Math. J. 60 (2010) 133–148. [11] J. Cao, W.B. Moors, A survey of topological games and their applications in analysis, RACSAM, Rev. R. Acad. Cienc. Exactas Fís. Nat. Ser. A Mat. 100 (2006) 39–49. [12] M.M. Choban, P.S. Kenderov, W.B. Moors, Pseudo-compact semi-topological groups, in: Proceedings of the Forty First Spring Conference of the Union of Bulgarian Mathematicians, Borovetz, April 2012, pp. 9–12. [13] R. Ellis, Locally compact transformation groups, Duke Math. J. 24 (1957) 119–125. [14] R. Ellis, A note on the continuity of the inverse, Proc. Am. Math. Soc. 8 (1957) 372–373. [15] Z. Frolík, Remarks concerning the invariance of Baire spaces under mappings, Czechoslov. Math. J. 11 (1961) 381–385. [16] S. Kempisty, Sur les fonctions quasi-continues, Fundam. Math. 19 (1932) 184–197. [17] P.S. Kenderov, I. Kortezov, W.B. Moors, Topological games and topological groups, Topol. Appl. 109 (2001) 157–165. [18] P.S. Kenderov, W.B. Moors, Fragmentability of groups and metric-valued function spaces, Topol. Appl. 159 (2012) 183–193. [19] P. Lin, W.B. Moors, Rich families, W -spaces and the product of Baire spaces, Math. Balk. (N.S.) 22 (2008) 175–187. [20] A.K. Mirmostafaee, Continuity of separately continuous mappings, Math. Slovak. 195 (2012) 1–7. [21] W.B. Moors, Any semitopological group that is homeomorphic to a product of Čech-complete spaces is a topological group, Set-Valued Var. Anal. (2013), in press. [22] D. Montgomery, Continuity in topological groups, Bull. Am. Math. Soc. 42 (1936) 879–882. [23] I. Namioka, Separate and joint continuity, Pac. J. Math. 51 (1974) 515–531. [24] H. Pfister, Continuity of the inverse, Proc. Am. Math. Soc. 95 (1985) 312–314. [25] Z. Piotrowski, Quasi-continuity and product spaces, in: Proc. Int. Conf. on Geometric Topology, Warszawa, 1978, PWN, Warsaw, 1980, pp. 349–352. [26] Z. Piotrowski, Separate and joint continuity, Real Anal. Exch. 11 (1985/1986) 293–322. [27] Z. Piotrowski, Separate and joint continuity II, Real Anal. Exch. 15 (1990) 248–258. [28] E.A. Reznichenko, Extension of functions defined on products of pseudocompact spaces and continuity of the inverse in pseudocompact groups, Topol. Appl. 59 (1994) 233–244. [29] J. Saint Raymond, Jeux topologiques et espaces de Namioka, Proc. Am. Math. Soc. 87 (1983) 499–504. [30] M.G. Tkachenko, Paratopological and semitopological vs topological group, in: K.P. Hart, J. van Mill, P. Simon (Eds.), Recent Progress in General Topology III, Atlantis Press/Springer, Paris, 2013, in press. [31] A.D. Wallace, The structure of topological semigroups, Bull. Am. Math. Soc. 61 (1955) 95–112. [32] W. Zelazko, A theorem on B0 division algebras, Bull. Acad. Pol. Sci. 8 (1960) 373–375.
Contents lists available at ScienceDirect
Topology and its Applications www.elsevier.com/locate/topol
Semitopological groups, Bouziad spaces and topological groups Warren B. Moors ∗ Department of Mathematics, The University of Auckland, Private Bag 92019, Auckland, New Zealand
a r t i c l e
i n f o
Article history: Received 24 February 2013 Received in revised form 14 August 2013 Accepted 15 August 2013
a b s t r a c t A semitopological group (topological group) is a group endowed with a topology for which multiplication is separately continuous (multiplication is jointly continuous and inversion is continuous). In this paper we use topological games to show that many semitopological groups are in fact topological groups. © 2013 Elsevier B.V. All rights reserved.
MSC: primary 22A20, 91A44 secondary 54E18, 54H11, 54H15 Keywords: Topological group Semitopological group Topological games Strongly Baire spaces
1. Introduction A semitopological group (topological group) is a group endowed with a topology for which multiplication is separately continuous (multiplication is jointly continuous and inversion is continuous). Recall that a function f : X × Y → Z that maps from a product of topological spaces X and Y into a topological space Z is said to be jointly continuous at a point (x, y) ∈ X × Y if for each neighbourhood W of f (x, y) there exists a pair of neighbourhood U of x and V of y such that f (U × V ) ⊆ W . If f is jointly continuous at each point of X × Y then we say that f is jointly continuous on X × Y . A related but weaker notion of continuity is the following. A function g : X × Y → Z that maps from a product of topological spaces X and Y into a topological space Z is said to be separately continuous on X × Y if for each x0 ∈ X and y0 ∈ Y the functions y → g(x0 , y) and x → g(x, y0 ) are both continuous on Y and X respectively. Ever since [22] there has been continued interest in determining topological properties of a semitopological group that are sufficient to ensure that it is a topological group. There have been many significant contributions to this area, see [1–10,12–14,17,18,20,22–32] to name but a few. Just about all of these results require the semitopological group to be regular (i.e., every closed subset and every point outside of this set, * Tel.: +64 9 3737 599x84746. E-mail address: [email protected]. 0166-8641/$ – see front matter © 2013 Elsevier B.V. All rights reserved. http://dx.doi.org/10.1016/j.topol.2013.08.008
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can be separated by disjoint open sets) and Baire (i.e., the intersection of any countable family of dense open sets is dense), and satisfy some additional completeness properties. This paper is no exception. However, in this paper we also require some additional notions from topology and game theory. We shall say that a subset Y of a topological space (X, τ ) is bounded in X if for each decreasing sequence of open sets (Un )n∈N in X, n∈N Un = ∅, whenever Un ∩ Y = ∅ for each n ∈ N. The game that we shall consider involves two players which we will call α and β. The “field/court” that the game is played on is a fixed topological space (X, τ ) with a fixed dense subset D. The name of the game is the GB (D)-game. After naming the game we need to describe how to “play” the GB (D)-game. The player labeled β starts the game every time (life is not always fair). For his/her first move the player β must select a pair (B1 , b1 ) consisting of a nonempty open subset B1 ⊆ X and a point b1 ∈ D. Next, α gets a turn. For α’s first move he/she must select a nonempty open subset A1 of B1 . This ends the first round of the game. In the second round, β goes first again and selects a pair (B2 , b2 ) consisting of a nonempty open subset B2 ⊆ A1 and a point b2 ∈ A1 ∩ D. Player α then gets to respond by choosing a nonempty open subset A2 of B2 . This ends the second round of the game. In general, after α and β have played the first n-rounds of the GB (D)-game, β will have selected pairs (B1 , b1 ), (B2 , b2 ), . . . , (Bn , bn ) consisting of nonempty open sets B1 , B2 , . . . , Bn and points b1 , b2 , . . . , bn in D and α will have selected nonempty open subsets A1 , A2 , . . . , An such that An ⊆ Bn ⊆ An−1 ⊆ Bn−1 ⊆ · · · ⊆ A2 ⊆ B2 ⊆ A1 ⊆ B1 , and bk+1 ∈ Ak ∩ D for all 1 k < n. At the start of the (n + 1)-round of the game, β goes first (again!) and selects a pair (Bn+1 , bn+1 ) consisting of a nonempty open subset Bn+1 of An and a point bn+1 ∈ An ∩ D. As with the previous n-rounds, the player α gets to respond to this move by selecting a nonempty open subset An+1 of Bn+1 . Continuing this procedure indefinitely (i.e., continuing on forever) the players α and β produce an infinite sequence (An , (Bn , bn ))n∈N called a play of the GB (D)-game. A partial play ((Ak , (Bk , bk )): 1 k n) of the GB (D)-game consists of the first n-moves of the GB (D)-game. As with any game, we need to specify a rule to determine who wins (otherwise, it is a very boring game). We shall declare that α wins a play (An , (Bn , bn ))n∈N of the GB (D)-game if: for each decreasing sequence / Un } is finite, for every n ∈ N. (Un )n∈N of open subsets of X, n∈N Un = ∅, whenever {k ∈ N: bk ∈ If α does not win a play of the GB (D)-game then we declare that β wins that play of the GB (D)-game. So every play is won by either α or β and no play is won by both players. Note that if α wins a play (An , (Bn , bn ))n∈N of the GB (D)-game then n∈N An = ∅. Continuing further into game theory we need to introduce the notion of a strategy. By a strategy t for the player β we mean a ‘rule’ that specifies each move of the player β in every possible situation. More precisely, a strategy t := (tn : n ∈ N) for β is an inductively defined sequence of τ × D-valued functions. The domain of t1 is the sequence of length zero, denoted by ∅. That is, Dom(t1 ) = {∅} and t1 (∅) ∈ (τ \ {∅}) × D. If t1 , t2 , . . . , tk have been defined then the domain of tk+1 is:
k (A1 , A2 , . . . , Ak ) ∈ τ \ {∅} : (A1 , A2 , . . . , Ak−1 ) ∈ Dom(tk ) and Ak ⊆ Bk , where (Bk , bk ) := tk (A1 , A2 , . . . , Ak−1 ) .
For each (A1 , A2 , . . . , Ak ) ∈ Dom(tk+1 ), tk+1 (A1 , A2 , . . . , Ak ) := (Bk+1 , bk+1 ) ∈ (τ \ {∅}) × D is defined so that Bk+1 ⊆ Ak and bk+1 ∈ Ak ∩ D. A partial t-play is a finite sequence (A1 , A2 , . . . , An−1 ) such that (A1 , A2 , . . . , An−1 ) ∈ Dom(tn ). A t-play is an infinite sequence (An )n∈N such that for each n ∈ N, (A1 , A2 , . . . , An−1 ) is a partial t-play. A strategy t := (tn : n ∈ N) for the player β is called a winning strategy if each play of the form: (An , tn (A1 , . . . , An−1 ))n∈N is won by β. We will call a topological space (X, τ ) a Bouziad space if it is
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regular and there exists a dense subset D of X such that the player β does not have a winning strategy in the GB (D)-game played on X. It follows from [29, Theorem 1] that each Bouziad space is in fact a Baire space. In this paper we will also be interested in another closely related game. This game, denoted GSB (D), is identical to the GB (D)-game, except for the definition of a win. In the GSB (D)-game we say that α wins a play (An , (Bn , bn ))n∈N of the GSB (D)-game if (i) there exists a subspace S of X and a subsequence (bnk )k∈N of (bn )n∈N , contained in S, such that for each decreasing sequence (Un )n∈N of open subsets of S, S / Un } is finite, for every n ∈ N and (ii) every subspace of Cp (S), n∈N Un = ∅, whenever {k ∈ N: bnk ∈ that is bounded in Cp (S), has a compact closure. We will call a topological space (X, τ ) a strong Bouziad space if it is completely regular and there exists a dense subset D of X such that the player β does not have a winning strategy in the GSB (D)-game played on X. It is easy to show that every strongly Bouziad space is a Bouziad space, and hence a Baire space. On the other hand, if X is completely regular and has the property that every subset of Cp (X), that is bounded in Cp (X), has a compact closure, then X is a Bouziad space if, and only if, it is a strongly Bouziad space. 2. Feeble continuity of multiplication Lemma 1. Let (G, ·, τ ) be a semitopological group. If (G, τ ) is a Bouziad space then for each pair of open neighbourhoods U and W of identity element e ∈ G there exists a nonempty open subset V of U such that V −1 ⊆ W · W · W . Proof. Suppose, in order to obtain a contradiction, that there exists a pair of open neighbourhoods U and W of e ∈ G such that for each nonempty open subset V of U , V −1 W · W · W . From this it follows that for each nonempty open subset V of U and each dense subset D of V there exists a point x ∈ V ∩ D such / W · W , because otherwise, that x−1 ∈ −1 −1 V −1 ⊆ V ∩ D ⊆ W · V ∩ D ⊆ W · W · W ⊆ W · W · W.
Recall that for any nonempty subset A of a semitopological group (H, ·, τ ) and any open neighbourhood W of the identity element e ∈ H, (A)−1 ⊆ W · A−1 . Now, let D be any dense subset of G such that β does not have a winning strategy in the GB (D)-game played on G. We will define a (necessarily non-winning) strategy t := (tn : n ∈ N) for β in the GB (D)-game played on G, but first we set, for notational reasons, A0 := U and b0 := e. −1 Step 1. Choose b1 ∈ A0 ∩D so that (b−1 = b−1 / W · W . Then choose U1 to be any open neighbourhood 0 ·b1 ) 1 ∈ of e, contained in U ∩ W , such that b1 · U1 ⊆ A0 . Then define t1 (∅) := (b1 · U1 , b1 ). Now, suppose that bj , Uj and tj (A1 , . . . , Aj−1 ) have been defined for each 1 j n so that: −1 (i) bj ∈ Aj−1 ∩ D and (b−1 ∈ / W · W; j−1 · bj ) (ii) Uj is an open neighbourhood of e, contained in U ∩ W , such that bj · Uj ⊆ Aj−1 ; (iii) tj (A1 , . . . , Aj−1 ) := (bj · Uj , bj ).
Step n + 1. Suppose that An is a nonempty open subset of bn · Un . That is, suppose that An is the n-th −1 move of the player α. Choose bn+1 ∈ An ∩ D so that (b−1 ∈ / W · W . Note that this is possible since n · bn+1 ) −1 −1 bn · (An ∩ D) is a dense subset of bn · An and −1 b−1 n · An ⊆ bn · (bn · Un ) = Un ⊆ U.
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Then choose Un+1 to be any neighbourhood of e, contained in U ∩ W , such that bn+1 · Un+1 ⊆ An . Finally, define tn+1 (A1 , . . . , An ) := (bn+1 · Un+1 , bn+1 ). Note that: −1 (i) bn+1 ∈ An ∩ D and (b−1 ∈ / W · W; n · bn+1 ) (ii) Un+1 is an open neighbourhood of e, contained in U ∩ W , such that bn+1 · Un+1 ⊆ An ; (iii) tn+1 (A1 , . . . , An ) := (bn+1 · Un+1 , bn+1 ).
This completes the definition of t. Since t is not a winning strategy for β there exists a play (An , tn (A1 , . . . , An−1 ))n∈N where α wins. Now there exists a k ∈ N such that bk ∈ ( n∈N Bn ) · W , because otherwise,
{bn : k + 1 n} ⊆ Bk \ Bn · W
for all k ∈ N.
n∈N
In which case,
∅ = Bk \ Bn · W ⊆ G \ Bn · W, k∈N
n∈N
n∈N
but on the other hand,
Bk \ Bn · W ⊆ Bn = Bn ⊆ Bn · W, n∈N
k∈N
n∈N
n∈N
n∈N
which is impossible. Therefore, we may indeed choose k ∈ N so that bk ∈
Bn
· W ⊆ Ak+1 · W ⊆ bk+1 · Uk+1 · W ⊆ bk+1 · W · W = bk+1 · W · W .
n∈N −1 Therefore, (b−1 = b−1 k · bk+1 ) k+1 · bk ∈ W · W . However, this contradicts the way bk+1 was chosen. This completes the proof. 2
If f : (X, τ ) → (Y, τ ) is a surjection acting between topological spaces (X, τ ) and (Y, τ ) then we say that f is feebly continuous on X if for each nonempty open subset V of Y , int[f −1 (V )] = ∅, [10,15]. Proposition 1. Let (G, ·, τ ) be a semitopological group. If multiplication, (h, g) → h · g, is feebly continuous on G × G then for each nonempty open subset U of G and n ∈ N there exist a point x in U and an open neighbourhood V of the identity element e ∈ G such that: x · V · V · V · · · V ⊆ U n-times
and
V · V · V · · · V · x ⊆ U. n-times
Proof. The proof of this follows from a simple induction argument and the fact that for each g ∈ G, both {g · U : U is a neighbourhood of e} and {U · g: U is a neighbourhood of e} are local bases for τ at the point g ∈ G. 2 Lemma 2. Let (G, ·, τ ) be a semitopological group. If (G, τ ) is a Bouziad space and the multiplication operation (h, g) → h · g is feebly continuous on G × G then the multiplication operation, (h, g) → h · g, is jointly continuous on G × G.
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Proof. Since (G, ·, τ ) is a semitopological group it is sufficient to show that multiplication is jointly continuous at (e, e). So, in order to obtain a contradiction, we will assume that multiplication is not jointly continuous at (e, e). Therefore, by the regularity of (G, τ ), there exists an open neighbourhood W of e so that for every neighbourhood U of e, U · U W . Since (G, τ ) is a Bouziad space there exists a dense subset DG of G such that β does not possess a winning strategy in the GB (DG )-game played on G. We will now inductively define a (necessarily non-winning) strategy t := (tn : n ∈ N) for the player β in the GB (DG )-game played on G. Step 1. We may choose a point x ∈ A0 := G and an open neighbourhood U of e ∈ G such that x · U ⊆ x · U · U · U ⊆ x · U · U · U ⊆ A0 . Next, we may pick y, z ∈ U such that y ·z ∈ / W (i.e., y ∈ / W · z −1 and so U \ (W · z −1 ) = ∅). By Proposition 1 −1 we may select a point y ∈ U \ (W · z ) and an open neighbourhood V of e, contained in U , such that V · V · V · V · y ⊆ U \ W · z −1 .
Then, (V ·V ·V ·x−1 )·(x·V )·y ·z ∩ W = ∅ and so (V · V · V ·x−1 )·(x·V )·y ·z ∩W = ∅. By Lemma 1 there exists a nonempty open subset B1 of x · V ⊆ x · U ⊆ x · U · U · U ⊆ A0 such that (B1 )−1 ⊆ V · V · V · x−1 . Thus, (B1 )−1 · B1 · y · z ∩ W = ∅. Choose b1 ∈ B1 · y · z ∩ DG ⊆ B1 · U · U ⊆ x · V · U · U ⊆ x · U · U · U ⊆ A0 .
Then define t1 (∅) := (B1 , b1 ) and U1 := B1 · y · z. Note that: (B1 )−1 · U1 ∩ W = ∅ — (∗1 ). Now suppose that (Bj , bj ), Uj and tj (A1 , . . . , Aj−1 ) have been defined for each 1 j n so that: (i) Uj is an open subset of Aj−1 and bj ∈ Uj ∩ DG ; (ii) Bj ⊆ Aj−1 and (Bj )−1 · Uj ∩ W = ∅ — (∗j ); (iii) tj (A1 , . . . , Aj−1 ) := (Bj , bj ). Step n + 1. Suppose that An is a nonempty open subset of Bn . That is, suppose that An is the n-th move of the player α. By Proposition 1 we may choose a point x ∈ An and an open neighbourhood U of e ∈ G such that x · U ⊆ x · U · U · U ⊆ x · U · U · U ⊆ An . Next, we may pick y, z ∈ U such that y · z ∈ / W (i.e., y ∈ / W · z −1 and so U \ (W · z −1 ) = ∅). Again by −1 Proposition 1 we may select a point y ∈ U \ (W · z ) and an open neighbourhood V of e, contained in U , such that V · V · V · V · y ⊆ U \ W · z −1 .
Then, (V · V · V · x−1 ) · (x · V ) · y · z ∩ W = ∅ and so (V · V · V · x−1 ) · (x · V ) · y · z ∩ W = ∅. By Lemma 1 there exists a nonempty open subset Bn+1 of x · V ⊆ x · U ⊆ An such that (Bn+1 )−1 ⊆ V · V · V · x−1 . Thus, (Bn+1 )−1 · Bn+1 · y · z ∩ W = ∅. Choose bn+1 ∈ Bn+1 · y · z ∩ DG ⊆ Bn+1 · U · U ⊆ x · V · U · U ⊆ x · U · U · U ⊆ An .
Then define tn+1 (A1 , . . . , An ) := (Bn+1 , bn+1 ) and Un+1 := Bn+1 · y · z. Note that: Bn+1 ⊆ An and (Bn+1 )−1 · Un+1 ∩ W = ∅ — (∗n+1 ).
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This completes the definition of t. Since t is not a winning strategy for β there exists a play (An , tn (A1 , . . . , An−1 ))n∈N where α wins. Note that since {bn : n k} ⊆ ( nk Un ) for all k ∈ N, ∅ =
k∈N
Un
⊆
nk
k∈N
Ak−1 =
k∈N
Ak =
k∈N
Bk =
Bn .
n∈N
Let b∞ ∈ k∈N ( nk Un ). Fix n ∈ N, then by Eq. (∗n ), b−1 ∞ ·Un ∩W = ∅, or equivalently, Un ∩(b∞ ·W ) = ∅. Hence, k∈N ( nk Un ) ∩ (b∞ · W ) = ∅. However, this contradicts the fact that b∞ ∈ k∈N ( nk Un ) and W is an open neighbourhood of e. Hence the multiplication operation on G is jointly continuous. 2
If f : (X, τ ) → (Y, τ ) is a function acting between topological spaces (X, τ ) and (Y, τ ) and x ∈ X then we say that f is quasi-continuous at x if for each neighbourhood W of f (x) and neighbourhood U of x there exists a nonempty open subset V ⊆ U such that f (V ) ⊆ W , [16]. Theorem 1. If (G, ·, τ ) is a semitopological group and (G, τ ) is a Bouziad space then (G, ·, τ ) is a topological group if (and only if ) the multiplication operation (h, g) → h · g is feebly continuous on G × G. Proof. From Lemma 2 we know that the multiplication operation on G is jointly continuous. Therefore, by Lemma 1, we see that inversion is quasi-continuous at e. The result now follows from [17, Lemma 4] where it is shown that each semitopological group with jointly continuous multiplication and inversion that is quasi-continuous at the identity element is a topological group. 2 Hence in the realm of Bouziad spaces, the problem of determining when a semitopological group is a topological group reduces to the problem of determining when a semitopological group has feebly continuous multiplication. 3. Strong quasi-continuity of multiplication Let X, Y and Z be topological spaces. We will say that a function f : X × Y → Z is strongly quasicontinuous, with respect to the second variable at (x, y) ∈ X × Y , if for each neighbourhood W of f (x, y) and each neighbourhood U of x there exists a nonempty open subset U ⊆ U and a neighbourhood V of y such that f (U × V ) ⊆ W , [25]. Remark 1. It follows from Proposition 1 that the multiplication operation on a semitopological group (G, ·, τ ) is feebly continuous on G × G if, and only if, it is strongly quasi-continuous, with respect to the second variable, at the point (e, e) ∈ G × G. If (X, τ ) is a topological space and a ∈ X then we shall denote by N (a) the family of all neighbourhoods of a. For any point a in a topological space (X, τ ) and any dense subset D of X we can consider the following two player topological game, called the Gp (a, D)-game. To define this game we must first specify the “rules” and then also specify the definition of a “win”. The moves of the player α are simple. He/she must always select a neighbourhood of the point a. However, the moves of the player β may depend upon the previous move of α. Specifically, for his/her first move β may select any point b1 ∈ D. For α’s first move, as mentioned earlier, α must select a neighbourhood A1 of a. Now, for β’s second move he/she must select a point b2 ∈ A1 ∩ D. For α’s second move he/she is entitled to select any neighbourhood A2 of a. In general, if α has chosen An ∈ N (a) as his/her n-th move of the Gp (a, D)-game then β is obliged to choose a point bn+1 ∈ An ∩ D. The response of α is then simply to
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choose any neighbourhood An+1 of a. Continuing in this fashion indefinitely, the players α and β produce a sequence ((bn , An ))n∈N of ordered pairs with bn+1 ∈ An ∩ D ⊆ An ∈ N (a) for all n ∈ N, called a play of the Gp (a, D)-game. A partial play ((bk , Ak ): 1 k n) of the Gp (a, D)-game consists of the first n moves of a play of the Gp (a, D)-game. We shall declare α the winner of a play ((bn , An ))n∈N of the Gp (a, D)-game if: for each decreasing sequence (Un )n∈N of open subsets of X, n∈N Un = ∅, whenever {k ∈ N: bk ∈ / Un } is finite, for every n ∈ N, otherwise, β is declared the winner. Note that if n∈N {bk : k n} = ∅ then α wins the corresponding play ((bn , An ))n∈N . A strategy for the player α is a rule that specifies his/her moves in every possible situation that can occur. More precisely, a strategy for α is an inductively defined sequence of functions t := (tn : n ∈ N). The domain of t1 is D1 and for each (b1 ) ∈ D1 , t1 (b1 ) ∈ N (a), i.e., ((b1 , t1 (b1 ))) is a partial play. Inductively, if t1 , t2 , . . . , tn have been defined then the domain of tn+1 is defined to be
(b1 , b2 , . . . , bn+1 ) ∈ Dn+1 : (b1 , b2 , . . . , bn ) ∈ Dom(tn ) and bn+1 ∈ tn (b1 , b2 , . . . , bn ) ∩ D .
For each (b1 , b2 , . . . , bn+1 ) ∈ Dom(tn+1 ), tn+1 (b1 , b2 , . . . , bn+1 ) ∈ N (a). Equivalently, for each (b1 , b2 , . . . , bn+1 ) ∈ Dom(tn+1 ), ((bk , tk (b1 , . . . , bk )): 1 k n + 1) is a partial play. A partial t-play is a finite sequence (b1 , b2 , . . . , bn ) ∈ Dn such that (b1 , b2 , . . . , bn ) ∈ Dom(tn ) or, equivalently, if bk+1 ∈ tk (b1 , b2 , . . . , bk ) ∩ D for all 1 k < n. A t-play is an infinite sequence (bn )n∈N such that for each n ∈ N, (b1 , b2 , . . . , bn ) is a partial t-play. A strategy t := (tn : n ∈ N) for the player α is said to be a winning strategy if each play of the form: (bn , tn (b1 , b2 , . . . , bn ))n∈N is won by α. ∗ We shall call a point a a nearly qD -point if the player α has a winning strategy in the Gp (a, D)-game played on X. For more information on topological games, see [11,19]. ∗ Lemma 3. Suppose that D is a dense subset of a topological space (X, τ ) and a is a nearly qD -point of X, then the player α possesses a strategy s := (sn : n ∈ N) in the Gp (a, D)-game such that every s-play is bounded in X.
Proof. Let t := (tn : n ∈ N) be a winning strategy for the player α in the Gp (a, D)-game. We shall inductively define a new strategy s := (sn : n ∈ N) for the player α. Step 1. Let (b1 ) be a partial s-play, i.e., b1 ∈ D, define s1 (b1 ) := t1 (b1 ). Now, suppose that s1 , s2 , . . . , sk have been defined so that for each 1 i k: (i) if (bn1 , . . . , bnm ) is a subsequence of (b1 , . . . , bi ) ∈ Dom(si ), then (bn1 , . . . , bnm ) ∈ Dom(tm ); (ii) if (b1 , . . . , bi ) ∈ Dom(si ) then si (b1 , . . . , bi ) =
tm (bn1 , . . . , bnm ): (bn1 , . . . , bnm ) is a subsequence of (b1 , . . . , bi ) .
Step k + 1. Suppose that (b1 , . . . , bk+1 ) is a partial s-play, i.e., (b1 , . . . , bk ) ∈ Dom(sk ) and bk+1 ∈ sk (b1 , . . . , bk ) ∩ D. Let (bn1 , . . . , bnm ) be any subsequence of (b1 , . . . , bk+1 ) and consider the following three cases: (i) if nm < k + 1, then (bn1 , . . . , bnm ) is a subsequence of (b1 , . . . , bk ) and so by the induction hypothesis (bn1 , . . . , bnm ) ∈ Dom(tm ); (ii) if nm = k + 1 and m > 1, then (bn1 , . . . , bnm−1 ) is a subsequence of (b1 , . . . , bk ) and so by the induction hypothesis (bn1 , . . . , bnm−1 ) ∈ Dom(tm−1 ). On the other hand, bnm = bk+1 ∈ sk (b1 , . . . , bk ) ∩ D ⊆ tm−1 (bn1 , . . . , bnm−1 ) ∩ D and so (bn1 , . . . , bnm ) ∈ Dom(tm ); (iii) if nm = k + 1 and m = 1, i.e., if n1 = k + 1 then (bn1 ) = (bk+1 ) ∈ Dom(t1 ).
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In all three cases, (bn1 , . . . , bnm ) ∈ Dom(tm ). Define sk+1 (b1 , . . . , bk+1 ) :=
tm (bn1 , . . . , bnm ): (bn1 , . . . , bnm ) is a subsequence of (b1 , . . . , bk+1 ) .
This completes the definition of s. Now suppose that (bn )n∈N is an s-play, we need to show that {bn : n ∈ N} is bounded in X. To this end, let (Un )n∈N be a decreasing sequence of open sets such that Un ∩{bk : k ∈ N} = ∅ for each n ∈ N. If for some n ∈ N, {k ∈ N: bk ∈ Un } is finite then n∈N Un = ∅. So we may suppose that for each n ∈ N, {k ∈ N: bk ∈ Un } is infinite. For each n ∈ N, let Jn := {k ∈ N: bk ∈ Un }. Then (Jn )n∈N is a decreasing sequence of infinite subsets of N. Hence there exists a strictly increasing sequence (nk )k∈N of the natural numbers such that nk ∈ Jk for all k ∈ N. We claim that (bnk )k∈N is a t-play. To show this we must show that for each m ∈ N, (bn1 , . . . , bnm ) ∈ Dom(tm ). However, (bn1 , . . . , bnm ) is a subsequence of (b1 , b2 , . . . , bnm ) ∈ Dom(snm ). Therefore, by the construction of the strategy s, (bn1 , . . . , bnm ) ∈ Dom(tm ). Hence (bnk )k∈N is a t-play. Now, |{k ∈ N: bnk ∈ / Un }| < n for each n ∈ N, therefore, n∈N Un = ∅. This shows that {bk : k ∈ N} is bounded in X. 2 Variations of the following result are well known, see [6,7,12,17,20]. Lemma 4. Let X be a strongly Bouziad space, Y be a topological space and Z be a completely regular space. If f : X × Y → Z is a separately continuous function and D is a dense subset of Y , then for each nearly ∗ qD -point y ∗ ∈ Y the function f is strongly quasi-continuous, with respect to the second variable, at each point of X × {y ∗ }. Proof. Let DX be any dense subset of X such that β does not have a winning strategy in the GSB (DX )-game played on X. (Note: such a dense subset is guaranteed by the fact that X is a strong Bouziad space.) We need to show that f is strongly quasi-continuous, with respect to the second variable, at each point (x∗ , y ∗ ) ∈ X × {y ∗ }. So in order to obtain a contradiction let us assume that f is not strongly quasi-continuous, with respect to the second variable, at some point (x∗ , y ∗ ) ∈ X × {y ∗ }. Then there exist open neighbourhoods W of f (x∗ , y ∗ ) and U of x∗ so that f (U × V ) W for each nonempty open subset U of U and each neighbourhood V of y0 . By the complete regularity of Z there exists a continuous function g : Z → [0, 1] such that g(f (x∗ , y ∗ )) = 1 and g(Z \ W ) = {0}. Let W := {z ∈ Z: g(z) > 3/4} ⊆ W . Note that by possibly making U smaller we may assume that f (x, y ∗ ) ∈ W for all x ∈ U . We will now inductively define a strategy t := (tn : n ∈ N) for the player β in the GSB (DX )-game played on X. However, we shall first: (a) let s := (sn : n ∈ N) be a strategy for the player α in the Gp (y ∗ , D)-game such that every s-play is bounded in Y . Note that by Lemma 3 such a strategy exists; (b) set (for notational reasons) x0 := x∗ , A0 := U and V0 := Y and let y0 be any element of D. Step 1. Select (x1 , y1 ) ∈ X × Y and an open set V1 and t1 (∅) so that: (i) y ∗ ∈ V1 := {y ∈ V0 ∩ s1 (y0 ): f (x0 , y) ∈ W }; (ii) (x1 , y1 ) ∈ (A0 ∩ DX ) × (V1 ∩ D) and f (x1 , y1 ) ∈ / W; (iii) t1 (∅) := (B1 , x1 ) where, B1 := {x ∈ A0 : f (x, y1 ) ∈ / W }. Now suppose that (xj , yj ), Vj and tj have been defined for each 1 j n so that for each 1 j n (i) y ∗ ∈ Vj := {y ∈ Vj−1 ∩ sj (y0 , . . . , yj−1 ): f (xj−1 , y) ∈ W }; (ii) (xj , yj ) ∈ (Aj−1 ∩ DX ) × (Vj ∩ D) and f (xj , yj ) ∈ / W; (iii) tj (A1 , . . . , Aj−1 ) := (Bj , xj ), where Bj := {x ∈ Aj−1 : f (x, yj ) ∈ / W }.
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Step n + 1. Suppose that An is a nonempty open subset of Bn . That is, suppose that An is the n-th move of the player α. Select (xn+1 , yn+1 ) ∈ X × Y and open set Vn+1 and tn+1 (A1 , . . . , An ) so that: (i) y ∗ ∈ Vn+1 := {y ∈ Vn ∩ sn+1 (y0 , . . . , yn ): f (xn , y) ∈ W }; (ii) (xn+1 , yn+1 ) ∈ (An ∩ DX ) × (Vn+1 ∩ D) and f (xn+1 , yn+1 ) ∈ / W; (iii) tn+1 (A1 , . . . , An ) := (Bn+1 , xn+1 ), where Bn+1 := {x ∈ An : f (x, yn+1 ) ∈ / W }. This completes the definition of t := (tn : n ∈ N). Now since t is not a winning strategy for the player β in the GSB (DX )-game there exists a play (An , tn (A1 , . . . , An−1 ))n∈N where α wins. Therefore, there exists a subspace S ⊆ X and a subsequence (xnk )k∈N of (xn )n∈N , contained in S, such that for each decreasing / Un } is finite, for every sequence (Un )n∈N of open subsets of S, n∈N Un S = ∅, whenever {k ∈ N: xnk ∈ n ∈ N. Define ϕ : Y → Cp (S) – [the continuous real-valued functions defined on S, endowed with the topology of pointwise convergence on S] by ϕ(y)(s) := (g ◦ f )(s, y)
for all s ∈ S.
Then ϕ is well defined and continuous on Y . Now, since (yn )n∈N is an s-play, {yn : n ∈ N} is bounded in Y and so {ϕ(ym ): m ∈ N} is bounded in Cp (S). Thus, by assumption, {ϕ(ym ): m ∈ N}τp is a compact subspace of Cp (S). Hence the sequence (ϕ(yn ))n∈N has a cluster point h ∈ C(S). Now, for each fixed k ∈ N, f (xnk , yi ) ∈ f {xnk } × Vi ⊆ f {xnk } × Vnk +1 ⊆ W
for all i > nk , since yi ∈ Vi for all i ∈ N. Therefore, ϕ(yi )(xnk ) ∈ (3/4, 1] for all i > nk and so h(xnk ) ∈ [3/4, 1] ⊆ (2/3, 1] for all k ∈ N. Since h is continuous, for every k ∈ N there exists a relatively open subset Uk of S such that xnk ∈ Uk ⊆ Ank −1 and h(Uk ) ⊆ (2/3, 1]. Hence the set
Ui S = ∅.
k∈N ik
[Here, X S denotes the closure of a subset X of S with respect to the relative topology on S.] Let x∞ ∈ k∈N ik Ui S ⊆ S. Then h(x∞ ) ∈ [2/3, 1]. On the other hand, if we again fix k ∈ N then f Ui × {yk } ⊆ f Ani −1 × {yk } ⊆ f Ai−1 × {yk } ⊆ f Ak × {yk } ⊆ Z \ W for all i > k. Therefore, f ( i>k Ui S × {yk }) ⊆ Z \ W for each k ∈ N and so f (x∞ , yk ) ∈ Z \ W for each k ∈ N; which implies that h(x∞ ) = 0. This however, contradicts our earlier conclusion that h(x∞ ) ∈ [2/3, 1]. Hence f is strongly quasi-continuous, with respect to the second variable, at (x∗ , y ∗ ). 2
Corollary 1. Suppose that (G, ·, τ ) is a semitopological group such that (G, τ ) is a strong Bouziad space. ∗ -point, for some dense subset Then (G, ·, τ ) is a topological group, provided (G, τ ) has at least one nearly qD D of G. Proof. Since for each g ∈ G, x → g · x is a homeomorphism on G, we may assume that the identity element ∗ -point, for some dense subset D of G. Then by Lemma 4 the multiplication operation e ∈ G is a nearly qD on G is strongly quasi-continuous, with respect to the second variable, at the point (e, e) ∈ G × G. Hence, by Remark 1 the multiplication operation on G is feebly continuous. The result now follows from Theorem 1 since every strongly Bouziad space is a Bouziad space. 2
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If (X, τ ) is a topological space and a ∈ X, then we call the point a, a q-point if there exists a sequence of neighbourhoods (Un )n∈N of a such that every sequence (xn )n∈N in X, with xn ∈ Un for all n ∈ N, has a cluster-point in X. Corollary 2. Suppose that (G, ·, τ ) is a semitopological group such that (G, τ ) is a Bouziad space. Then (G, ·, τ ) is a topological group, provided (G, τ ) has at least one q-point. Proof. It is well known that if K is a q-space, then every subset of Cp (K), that is bounded in Cp (K), has a compact closure. Hence, by the comment at the end of the introduction to this paper, we see that (G, τ ) is ∗ -point of G. Hence this a strong Bouziad space. Moreover, it is clear that every q-point of G is a nearly qG result follows from Corollary 1. 2 Corollary 3. ([28]) Suppose that (G, ·, τ ) is a semitopological group such that (G, τ ) is pseudo-compact. Then (G, ·, τ ) is a topological group, provided every subset of Cp (G), that is bounded in Cp (G), has a compact closure. Proof. From the comment at the end of the introduction to this paper, it follows that (G, τ ) is a strong ∗ -point of G. So the Bouziad space. Furthermore, since G is pseudo-compact, every point of G is a nearly qG result follows from Corollary 1. 2 Corollary 4. ([21]) If (G, ·, τ ) is a semitopological group such that (G, τ ) is homeomorphic to a product of Čech-complete spaces, then (G, ·, τ ) is a topological group. Proof. It is easy to see that every nearly strongly Baire space (see, [21] for the definition) is a strong Bouziad ∗ -point. Now, in [21] it is space and that every nearly qD -point (see [21] for the definition) is a nearly qD shown that every space that is homeomorphic to a product of Čech-complete spaces is a nearly strongly Baire space with at least one nearly qD -point. Hence the result follows from Corollary 1. 2 Corollary 5. ([17]) If (G, ·, τ ) is a semitopological group such that (G, τ ) is a completely regular strongly Baire space, then (G, ·, τ ) is a topological group. Proof. It is easy to see that every completely regular, strongly Baire space (see [17] for the definition) is a ∗ -point. The result then follows from Corollary 1. 2 strong Bouziad space and possesses at least one nearly qD Acknowledgement After this paper went to press the author became aware of the paper “Continuity of the inverse” located at: http://gtopology.math.msu.su/node/175 by Evgenii Reznichenko where the author proves a theorem similar to Theorem 1. References [1] A.V. Arhangel’skii, M.M. Choban, P.S. Kenderov, Topological games and topologies on groups, Math. Maced. 8 (2010) 1–19. [2] A.V. Arhangel’skii, M.M. Choban, P.S. Kenderov, Topological games and continuity of group operations, Topol. Appl. 157 (2010) 2542–2552. [3] A.V. Arhangel’skii, E.A. Reznichenko, Paratopological and semitopological groups versus topological groups, Topol. Appl. 151 (2005) 107–119. [4] A.V. Arhangel’skii, M.G. Tkachenko, Topological Groups and Related Structures, Atlantis Press, Amsterdam/Paris, 2008. [5] A. Bareche, A. Bouziad, Some results on separate and joint continuity, Topol. Appl. 157 (2010) 327–335. [6] A. Bouziad, The Ellis theorem and continuity in groups, Topol. Appl. 50 (1993) 73–80. [7] A. Bouziad, Continuity of separately continuous group actions in p-spaces, Topol. Appl. 71 (1996) 119–124.
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